Coulomb energy, remnant symmetry in Coulomb gauge, and phases of non-abelian gauge theories

1. Coulomb energy, remnant symmetry in Coulomb gauge, and phases of non-abelian gauge theories with J. Greensite and D. Zwanziger (a part with R. Bertle and M. Faber) hep-lat/0302018 (JG, ŠO) hep-lat/0309172 (JG, ŠO) hep-lat/0310057 (RB, MF, JG, ŠO) paper in preparation (JG, ŠO, DZ)

2. Confinement problem in QCD • The problem remains unsolved and lucrative: • The phenomenon attributed to field configurations with non-trivial topology: • Instantons? • Merons? • Abelian monopoles? • Center vortices? • Their role can be (and has been) investigated in lattice simulations.

3. Why Coulomb gauge? • Two features of confinement: • Long-range confining force between coloured quarks. • Absence of gluons in the particle spectrum. • Requirements on the gluon propagator at zero momentum: • A strong singularity as a manifestation of the long-range force. • Strongly suppressed because there are no massless gluons. • Difficult to reach simultaneously in covariant gauges! • In the Coulomb gauge: • Long-range force due to instantaneous static colour-Coulomb field. • The propagator of transverse, would-be physical gluons suppressed.

4. Confinement scenario in Coulomb gauge • hA0A0i propagator: • Classical Hamiltonian in CG:

5. Coulomb energy • Physical state in CG containing a static pair: • Correlator of two Wilson lines: • Then:

6. Measurement of the Coulomb energy on a lattice • Lattice Coulomb gauge: maximize • Wilson-line correlator: • Questions: • Does V(R,0) rise linearly with R at large b? • Does scoul match sasympt?

8. Center vortices and Coulomb energy

10. Scaling of the Coulomb string tension? • Saturation? No, overconfinement!

12. Center symmetry and confinement • Different phases of a stat. system are often characterized by the broken or unbroken realization of some global symmetry. • Polyakov loop not invariant: • On a finite lattice, below or above the transition, <P(x)>=0, but:

13. Coulomb energy and remnant symmetry • Maximizing R does not fix the gauge completely: • Under these transformations: • Both L and Tr[L] are non-invariant, their expectation values must vanish in the unbroken symmetry regime. • The confining phase is therefore a phase of unbroken remnant gauge symmetry; i.e. unbroken remnant symmetry is a necessary condition for confinement.

14. An order parameter for remnant symmetry in CG • Define • Order parameter (Marinari et al., 1993): • Relation to the Coulomb energy:

15. Massless phase: field spherically symmetric Compact QED, b>1 Confined phase: field collimated into a flux tube Compact QED, b<1 Pure SU(N) at low T SU(N)+adjoint Higgs Screened phases: Yukawa-like falloff of the field Pure SU(N) at high T SU(N)+adjoint Higgs SU(N)+matter field in fund. representation Different phases of gauge theories (ZN center symmetric)

16. Compact QED4

17. SU(2) gauge-adjoint Higgs theory

21. A surprise: SU(2) in the deconfined phase • Does remnant and center symmetry breaking always go together? NO!

23. Center vortices and Coulomb energy • Center vortices are identified by fixing to an adjoint gauge, and then projecting link variables to the ZN subgroup of SU(N). The excitations of the projected theory are known as P-vortices. • Direct maximal center gauge: • Vortex removal: • What happens when “vortex-removed” configurations are brought to the Coulomb gauge? • Coulomb energy

24. SU(2) in the deconfined phase: an explanation (?) • Spacelike links are a confining ensemble even in the deconfinement phase: spacelike Wilson loops have an area law behaviour. • Removing vortices removes the rise of the Coulomb potential. • Thin vortices lie on the Gribov horizon! (A proof: D. Zwanziger.)

25. SU(2) gauge-fundamental Higgs theory

26. SU(2) with fundamental Higgs

27. b=0

31. Kertész line?

32. Conclusions • The Coulomb string tension much larger than the true asymptotic string tension. • Confining property of the color Coulomb potential is tied to the unbroken realization of the remnant gauge symmetry in CG. • The deconfined phase in pure GT, and the “confinement” region of gauge-fundamental Higgs theory: color Coulomb potential is asymptotically linear, even though the static quark potential is screened. Center symmetry breaking, spontaneous or explicit, does not necessarily imply remnant symmetry breaking. • Strong correlation between the presence of center vortices and the existence of a confining Coulomb potential. Thin center vortices lie on the Gribov horizon. The transition between regions of broken/unbroken remnant symmetry: percolation transition (Kertész line).